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G = C2×C40⋊C2  order 160 = 25·5

Direct product of C2 and C40⋊C2

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C40⋊C2, C8⋊8D10, C4.6D20, C40⋊9C22, C10⋊1SD16, C20.29D4, C20.28C23, D20.6C22, C22.12D20, Dic10⋊3C22, (C2×C8)⋊5D5, (C2×C40)⋊7C2, C5⋊1(C2×SD16), C10.9(C2×D4), (C2×D20).5C2, C2.11(C2×D20), (C2×C10).16D4, (C2×C4).79D10, (C2×Dic10)⋊5C2, C4.26(C22×D5), (C2×C20).88C22, SmallGroup(160,123)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C20 — C2×C40⋊C2
Chief series
C1 — C5 — C10 — C20 — D20 — C2×D20 — C2×C40⋊C2
Lower central
C5 — C10 — C20 — C2×C40⋊C2
Upper central
C1 — C22 — C2×C4 — C2×C8

Generators and relations for C2×C40⋊C2
 G = < a,b,c | a2=b40=c2=1, ab=ba, ac=ca, cbc=b19 >

Subgroups: 280 in 68 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C2×C8, SD16, C2×D4, C2×Q8, Dic5, C20, D10, C2×C10, C2×SD16, C40, Dic10, Dic10, D20, D20, C2×Dic5, C2×C20, C22×D5, C40⋊C2, C2×C40, C2×Dic10, C2×D20, C2×C40⋊C2
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, D10, C2×SD16, D20, C22×D5, C40⋊C2, C2×D20, C2×C40⋊C2

Smallest permutation representation of C2×C40⋊C2
►On 80 points
Generators in S80
(1 47)(2 48)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 63)(18 64)(19 65)(20 66)(21 67)(22 68)(23 69)(24 70)(25 71)(26 72)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 79)(34 80)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)

(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)

(1 47)(2 66)(3 45)(4 64)(5 43)(6 62)(7 41)(8 60)(9 79)(10 58)(11 77)(12 56)(13 75)(14 54)(15 73)(16 52)(17 71)(18 50)(19 69)(20 48)(21 67)(22 46)(23 65)(24 44)(25 63)(26 42)(27 61)(28 80)(29 59)(30 78)(31 57)(32 76)(33 55)(34 74)(35 53)(36 72)(37 51)(38 70)(39 49)(40 68)
 
G:=sub<Sym(80)| (1,47)(2,48)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,67)(22,68)(23,69)(24,70)(25,71)(26,72)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,47)(2,66)(3,45)(4,64)(5,43)(6,62)(7,41)(8,60)(9,79)(10,58)(11,77)(12,56)(13,75)(14,54)(15,73)(16,52)(17,71)(18,50)(19,69)(20,48)(21,67)(22,46)(23,65)(24,44)(25,63)(26,42)(27,61)(28,80)(29,59)(30,78)(31,57)(32,76)(33,55)(34,74)(35,53)(36,72)(37,51)(38,70)(39,49)(40,68)>;
 
G:=Group( (1,47)(2,48)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,67)(22,68)(23,69)(24,70)(25,71)(26,72)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,47)(2,66)(3,45)(4,64)(5,43)(6,62)(7,41)(8,60)(9,79)(10,58)(11,77)(12,56)(13,75)(14,54)(15,73)(16,52)(17,71)(18,50)(19,69)(20,48)(21,67)(22,46)(23,65)(24,44)(25,63)(26,42)(27,61)(28,80)(29,59)(30,78)(31,57)(32,76)(33,55)(34,74)(35,53)(36,72)(37,51)(38,70)(39,49)(40,68) );
 
G=PermutationGroup([[(1,47),(2,48),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,55),(10,56),(11,57),(12,58),(13,59),(14,60),(15,61),(16,62),(17,63),(18,64),(19,65),(20,66),(21,67),(22,68),(23,69),(24,70),(25,71),(26,72),(27,73),(28,74),(29,75),(30,76),(31,77),(32,78),(33,79),(34,80),(35,41),(36,42),(37,43),(38,44),(39,45),(40,46)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(1,47),(2,66),(3,45),(4,64),(5,43),(6,62),(7,41),(8,60),(9,79),(10,58),(11,77),(12,56),(13,75),(14,54),(15,73),(16,52),(17,71),(18,50),(19,69),(20,48),(21,67),(22,46),(23,65),(24,44),(25,63),(26,42),(27,61),(28,80),(29,59),(30,78),(31,57),(32,76),(33,55),(34,74),(35,53),(36,72),(37,51),(38,70),(39,49),(40,68)]])
 

C2×C40⋊C2 is a maximal subgroup of
 C8⋊5D20  C8.8D20  C42.16D10  C8⋊D20  C8.D20  D20.31D4  D20.32D4  D20⋊14D4  Dic10⋊14D4  Dic10⋊2D4  D20.8D4  D4⋊3D20  D20.D4  Dic10.11D4  Q8⋊2D20  Q8.D20  Dic5⋊SD16  C20⋊SD16  D20.19D4  C42.36D10  Dic10⋊8D4  Dic5⋊8SD16  C8⋊8D20  C8⋊3D20  C40⋊21(C2×C4)  C8.24D20  C40⋊30D4  C40⋊2D4  D4.3D20  C40⋊11D4  C40.43D4  C40⋊15D4  C40.37D4  D4.11D20  C2×D5×SD16  D8⋊11D10
C2×C40⋊C2 is a maximal quotient of
 C40⋊9Q8  C20.14Q16  C8⋊5D20  C4.5D40  C23.34D20  D20.31D4  C23.38D20  Dic10⋊14D4  C20⋊SD16  D20⋊3Q8  Dic10⋊8D4  Dic10⋊4Q8  C40⋊30D4

46 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D5A5B8A8B8C8D10A···10F20A···20H40A···40P
order122222444455888810···1020···2040···40
size111120202220202222222···22···22···2

46 irreducible representations

dim11111222222222
type++++++++++++
imageC1C2C2C2C2D4D4D5SD16D10D10D20D20C40⋊C2
kernelC2×C40⋊C2C40⋊C2C2×C40C2×Dic10C2×D20C20C2×C10C2×C8C10C8C2×C4C4C22C2
# reps141111124424416

Matrix representation of C2×C40⋊C2 ►in GL3(𝔽41) generated by

4000
010
001
,
100
02728
01318
,
100
0134
0040
G:=sub<GL(3,GF(41))| [40,0,0,0,1,0,0,0,1],[1,0,0,0,27,13,0,28,18],[1,0,0,0,1,0,0,34,40] >;
 

C2×C40⋊C2 in GAP, Magma, Sage, TeX 

C_2\times C_{40}\rtimes C_2
 
% in TeX
 
G:=Group("C2xC40:C2");
 
// GroupNames label
 
G:=SmallGroup(160,123);
 
// by ID
 
G=gap.SmallGroup(160,123);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,218,50,579,69,4613]);
 
// Polycyclic
 
G:=Group<a,b,c|a^2=b^40=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^19>;
 
// generators/relations
 

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